INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

COMPUTATION OF RENAMEABLEHORN BACKDOORS

Stephan Kottler, Michael Kaufmann andCarsten Sinz

University of Tuebingen, Germany

15th May 2008 @ SAT’08 Guangzhou

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

OUTLINE

1 INTRODUCTION2 GRAPH APPROACH

Theoretical BackgroundGreedy HeuristicApproximation

3 EXPERIMENTSComparing ResultsSimplification of Dependency Graphs

4 CONCLUSION

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

INTRODUCTION

Backdoor Set: particular subset of variablesWilliams, Gomes and Selman 2003

Focus on variables of a strong backdoor issufficient to decide satisfiability

Example of real-world instance: 6,700 vars &440,000 clauses→ backdoor with 12 variables

Random instances have much larger backdoors(Interian)

Finding a minimum backdoor is hard (Szeider)

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

DELETION BACKDOORS

Introduced by Nishimura, Ragde and Szeider

Defined with respect to a base class C

C recognizable and solvable in poly. timee.g. base classes Horn, 2-SAT, Renameable Horn

B ⊆ V is a deletion backdoor if F −B belongs to CF −B: remove from F all occurrences of variables (pos./neg.) in B

A deletion backdoor is a strong backdoor if C isclause-induced

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

RECENT WORK

Paris et al.: RHorn-Backdoors for ZChaff

Try to rename variables to increase the number ofHorn clauses (WalkSat)Greedily choose variables for the backdoor

Dilkina, Gomes and Sabharwal:Computed optimal Backdoors for different baseclassesMinimum Renameable Horn Backdoors≤ Minimum Renameable Horn Deletion Backdoors

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

RHORN AS A GRAPH PROBLEM

For formula F we create a graph G = (V ,E):V: Each variable xi entails two vertices:

k0i represents that xi has to be renamedk1i represents that xi must not be renamed

. . . to make F HornE: Represent the implications of renaming

or not renaming variables (according to clauses)

EXAMPLE : (xi ∨xj ∨ . . .)If xj is renamed than xi has also to be renamedIf xi is not renamed than xj must not be renamed

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

CONFLICT LOOPS ANDCONFLICT SETSA variable xi has a conflict loop if there is a path from k0ito k1i and vice versa. The set of variables involved in aconflict loop is a conflict set.

A formula F is Renameable Horn iff there existsno variable that has a conflict loop in the graph.

Lewis proved: For any formula F there is a2-SAT fromula that is satisfiable iff F is RenameableHornThe Dependency Graph is the Implication Graph(Aspvall et al.) of Lewis’ 2-SAT-instance

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

REDUCTION LEMMASIMPLIFICATION OF THE GRAPH

If variable xi does not have a conflict loop thenneither vertex k0i nor vertex k

1i can be involved

in a conflict loop of any other variable.

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

RHORN DELETION BACKDOORS. . .AS A GRAPH PROBLEM

Goal: We have to get rid of all conflict loops!

Delete variables to remove conflict loops.

Why not using strongly connectedcomponents?⇒ At the beginning there is only one SCCTwo approaches to destroy conflict loops

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

A GREEDY HEURISTICTO DESTROY ALL CONFLICT LOOPS

Function greedyRHornBackdoor (F )G = (V ,E)← Dependency Graph of FS← computeConflictSets (G,V )B← /0 (start with an empty backdoor)while S 6= /0 do

xi ← choose variable according to heuristicB← B ∪{xi}delete vertices k0i ,k

1i and incident edges

U← variables, whose conflict loops were destroyedS← S ∪ computeConflictSets (G,U)Apply reduction rules according to Reduction Lemma

return B

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

APPROXIMATIONOF OPTIMAL RENAMEABLE HORN DELETION BACKDOORS

Function approxRHornBackdoor (F )G = (V ,E)← Dependency Graph of F ; B← /0while G contains at least one conflict loop do

C← vertices of one (preferably small) conflict loopB← B ∪{var(k) : k ∈ C}Hide all vertices related to vars in B (and incident edges)Apply reduction rules according to Reduction Lemma

forall x ∈ B doReinsert vertices (and edges) related to xif G contains no conflict loop then B← B \{x}else Undo reinsertion of vertices and edges related to x

return B

Inspired by an algorithm for the FEEDBACK ARC SET problem(Demetrescu & Finocchi)

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

COMPARISON. . .TO LOCAL SEARCH APPROACHES

Number of WalkSat Dependency GraphInstance Vars Cls (Paris) heuristic 2-phase

apex7_w5 1500 11695 740 663 3.93s 627 1.60sc499_w5 2070 22470 885 837 5.33s 818 2.35sdp10s10 8372 23004 2635 1449 26.90s 1498 2.08slisa20_2 1201 6563 820 774 0.87s 799 0.22srand_net40 2000 5921 811 665 2.79s 692 0.34svda_w9 6498 130997 4809 4488 6m 4293 5mvmpc_21 441 45339 439 437 1.50s 424 10.13svmpc_25 625 76755 603 610 5.48s 605 46.17s

!

Graph Approach is independent of the number ofrenamings!

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

SIMPLIFICATION OF THE GRAPHSFOR EASY INSTANCES

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

SIMPLIFICATION OF THE GRAPHSFOR HARD INSTANCES

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

PRACTICAL RELEVANCERESULTS FOR INSTANCES OFSAT COMPETITION 2007

Number of Heuristic ApproximationInstance Vars Cls BD % Time BD % Time

AProVE07-06 46335 632886 4485 9% 28.90s 4376 9% 12.24seq.a.braun.12 1694 5726 639 37% 36.59s 634 37% 1.07seq.a.braun.13 2010 6802 765 38% 45.17s 755 37% 1.86sdspam_vc1080 118298 375379 32018 27% 289m 40220 33% 78mmizh-md5-47-3 65604 273522 15077 22% 25m 16687 25% 1m

QG6-ukn2726 2123 9177 710 33% 15.47s 491 23% 30.18sbqwh.40.520 2211 14710 1431 64% 8.28s 1458 65% 0.64scontest02-26 744 2464 376 50% 0.13s 351 47% 0.08sgensys-ukn002 2129 8961 702 32% 18.26s 483 22% 29.01s

unif-k3-r4.25 450 1912 238 52% 0.76s 243 54% 0.28sunif-k7-r89 75 6675 74 98% 0.15s 74 98% 0.19sunif2p-p0.7 3500 9344 1065 30% 1m 1116 31% 1munif2p-p0.9 1170 4234 525 44% 3.87s 552 47% 3.32s

!

Industrial / Crafted / Random Instances

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INTRODUCTION GRAPH APPROACH EXPERIMENTS CONCLUSION

CONCLUSION

What we did . . .

Two Approaches to compute Renameable Horn(Deletion) Backdoors

Realistic for small but hard instances!

Open Problems

How can Backdoors be used for the solvingprocess?

Fast Computation of Non-Deletion Backdoors

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